Doraiswami Ramkrishna

On the solution of population balance equations by discretization—II. A moving pivot technique

Abstract A new framework for the discretization of continuous population balance equations (PBEs) is presented in this work. It proposes that the discrete equations for aggregation or breakage processes be internally consistent with regard to the desired moments of the distribution. Based on this framework, a numerical technique has been developed. It considers particle populations in discrete and contiguous size ranges to be concentrated at representative volumes. Particulate events leading to the formation of particle sizes other than the representative sizes are incorporated in the set of discrete equations such that properties corresponding to two moments of interest are exactly preserved. The technique presented here is applicable to binary or multiple breakage, aggregation, simultaneous breakage and aggregation, and can be adapted to predict the desired properties of an evolving size distribution more precisely. Existing approaches employ successively fine grids to improve the accuracy of the numerical results. However, a simple analysis of the aggregation process shows that significant errors are introduced due to steeply varying number densities across a size range. Therefore, a new strategy involving selective refinement of a relatively coarse grid while keeping the number of sections to a minimum, is demonstrated for one particular case. Furthermore, it has been found that the technique is quite general and yields excellent predictions in all cases. This technique is particularly useful for solving a large class of problems involving discrete-continuous PBEs such as polymerization-depolymerization, aerosol dynamics, etc.

On the solution of population balance equations by discretization - III. Nucleation, growth and aggregation of particles

Abstract A new discretization method for solving population balance equations for simultaneous nucleation, growth and aggregation of particles is proposed. The method combines the best features of our discretization technique (Kumar and Ramkrishna, 1996, Chem. Engng. Sci. 51 , 1311–1337), i.e., designing discrete equations to obtain desired properties of a size distribution directly, applicability to an arbitrary grid to control resolution and computational efficiency, with the method of characteristics to offer a technique which is very general, powerful and overcomes the crucial problems of numerical diffusion and stability that beset the previous techniques in this area. The proposed technique has been tested for pure growth, simultaneous growth and aggregation, and simultaneous nucleation and growth for a large number of combinations obtained by changing functions for nucleation rate, growth rate, aggregation kernel and initial condition. In all cases, the size distributions obtained from the proposed technique and those obtained analytically are in excellent agreement. The presence of moving discontinuities, which is unavoidable due to the hyperbolic nature of the governing equation, is addressed with no additional difficulty in all of the test problems.

Statistics and dynamics of procaryotic cell populations

Abstract The formulation of a mathematical theory of a cell population requires the ability to specify quantitatively the physiological state of individual cells of the population. In procaryotic cells (bacteria and blue-green algae), the intracellular structure is of a relatively simple nature, and it is postulated that the physiological state of such a cell is specified by its biochemical composition. If we postulate further that the growth rate of a cell and its fission probability depend only on the cells current physiological state and on the current state of the cells environment, then an equation of change for the distribution of physiological states in a population can be derived. In addition, an equation of change for the state of the cellular environment can be obtained. These equations allow us to predict the statistical and dynamical behavior of a cell population from information obtained by analysis of cellular and subcellular structure and function.

Theoretical investigations of dynamic behavior of isothermal continuous stirred tank biological reactors

Abstract The dynamic behavior of an isothermal biological CSTR modelled by idealized cell and substrate balance equations has been investigated theoretically in terms of multiplicity and stability of steady states and existence and stability character of limit cycles. Various types of dynamic behavior have been classified in terms of a Damkohler number and two other system parameters. The predicted types of behavior have been illustrated by numerical computation of cell and substrate concentration trajectories.

Population balance modeling. Promise for the future

Population balance modeling has received an unprecedented amount of attention during the past few years from both academic and industrial quarters because of its applicability to a wide variety of particulate processes. In this article, a fresh look is taken of the basic issues of the application of population balances towards strengthening the approach as well as widening the scope of their applications with regard to formulation, computational methods for solution, inverse problems, control of particle populations and stochastic modeling.

A model for transitional breakage probability of droplets in agitated lean liquid-liquid dispersions

Abstract A model for transitional breakage probability of droplets in agitated lean fiquid-liquid dispersions is proposed based on the mechanism of breakage of droplets due to their oscillations resulting from relative velocity fluctuations. A universal transitional breakage probability in terms of non-dimensionalized drop diameter is derived for all dispersed phases whose density and viscosity are almost the same as that of continuous phase. The maximum stable drop diameter ds derived from the model, shows a dependence of NWe−0.6. It is shown that a “power law” approximation Kvn is valid for transitional breakage probability for d/ds up to 2. The exponent 2.67, predicted by this model corresponds rather well with an estimate of 2, obtained from experimental observations. A functional relation for the rate constant K in terms of the parameters and physical properties of the system is derived. A universal non-dimensionalized equilibrium drop-size distribution for agitated lean liquid-liquid dispersions is derived by analytical solution of a population balance equation simplified by order of magnitude estimates. Interestingly enough, this analytical solution is the same as the Gaussian distribution suggested empirically by Chen and Middleman.

Metabolic Engineering from a Cybernetic Perspective. 1. Theoretical Preliminaries

The theoretical basis of a cybernetic metabolic network design and analysis framework, which has been subsequently successfully applied to predict system response to genetic alteration, is presented. This conceptual methodology consists of three main branches, namely, a model realization framework, a representation of genetic alteration, and lastly, a metabolic design component. These concepts are introduced as a series of postulates that describe the basic tenets of the approach. Each branch is discussed in turn, starting with the cybernetic representation of arbitrarily complex metabolic networks. A set of postulates is put forth that affords the modular construction of cybernetic models of metabolic networks using as a base a library of elementary pathways. This is followed by a discussion of the representation of genetic alterations within the cybernetic framework. It is postulated that the objective of the base network and the altered system are identical (at least on the time scale required for the organism to “learn” new objectives). This implies, with respect to resource allocation, that the base network and its genetically altered counterpart may still be treated as optimal systems; however, the set of competing physiological choices open to the altered network expands or contracts depending upon the nature of the genetic perturbation. Lastly, to add a predictive design aspect to the methodology, we present a set of postulates that outline the application of metabolic control analysis to cybernetic model systems. We postulate that sensitivity coefficients computed from a cybernetic model, although still local in scope, have the added benefit of a systematic representation of regulatory function as described by the cybernetic variables. Thus, information gained from sensitivity measurements stemming from a cybernetic model include the explicit input of metabolic regulation, a component that is lacking in a purely kinetic representation of metabolic function. The sensitivity results can then be employed to develop qualitative strategies for the rational alteration of metabolic function, which can be evaluated by simulation of an appropriately modified cybernetic model of the base network.

Cybernetic modeling of growth in mixed, substitutable substrate environments: Preferential and simultaneous utilization

Growth of microorganisms on substitutable substrate mixtures display diverse growth dynamics characterized by simultaneous or preferential uptake of carbon sources. This article shows that cybernetic modeling concepts which were successful in predicting diauxic growth patterns can be extended to describe simultaneous consumption of substrates. Thus the growth of Escherichia coli on mixtures of glucose and organic acids such as pyruvate, fumarate, and succinate has been described successfully by the cybernetic model presented here showing both diauxic and simultaneous uptake when observed. The model also describes the changes in utilization patterns that occur under changing dilution rates, substrate concentrations, and models of preculturing. The model recognizes the importance of the synthesis of biosynthetic precursors in cell growth through a kinetic structure that is quite general for any mixture of carbon-energy sources. (c) 1996 John Wiley & Sons, Inc.

Conjugated Graetz problems—I: General formalism and a class of solid-fluid problems

Abstract Conjugated Graetz problems arise where two (or more) phases, with at least one phase in fully developed flow, exchange energy or mass across an intervening surface. In these problems the temperature or concentration fields are coupled through the conjugating conditions which express the continuity of fluxes and the rate of transfer. A general formalism is presented first for the analysis of these problems, employing a matrix differential operator with respect to the radial variable and following the decomposition technique of [1–3]. The aforementioned operator is shown to be symmetric in its domain, possessing a denumerable set of eigenvalues and a complete set of eigenvectors. A class of solid-fluid problems, involving the removal of heat from a heated solid cylinder by a surrounding annular-flow fluid, is then discussed in detail; analytical solutions are obtained by expansion in terms of the above eigenvectors. These solid-fluid problems may be viewed as extensions of the extended, one-phase, Graetz problem with prescribed wall flux [2].

Droplet breakage in stirred dispersions. Breakage functions from experimental drop-size distributions

Transient breakage drop-size distributions have been experimentally measured using an image analysis technique. The transient distributions show self-similar behavior. The breakage rate and daughter-drop distribution functions have been determined using an inverse-problem approach which takes advantage of this self-similarity. The inverse-problem results show that the breakage rate is not a power law function of the drop size. The breakage rate is found to increase sharply with the drop size and the stirrer speed while decreasing sharply with increase in the interfacial tension. It is also found to decrease with increase in the dispersed phase viscosity, though the dependence on the viscosity is weaker than on the other variables. The daughter drop distribution was found to be relatively insensitive to the stirrer speed and interfacial tension, but was found to depend on the dispersed phase viscosity. As the drop viscosity increases, the breakage becomes more erosive in nature, leading to a broader size distribution of daughter drops. Generalized correlations for the breakage rate and daughter-drop distribution which account for the effect of physical properties and experimental conditions are presented. These relations will be very useful in predicting the drop-size distributions in stirred dispersions. Models for the breakage functions are compared with those determined in this study and the model predictions of the transient-size distributions are compared with the experimental data.